Speaker:Patrick BrosnanTitle: The essential dimension of an algebraic stackAbstract:
The essential dimension of algbebraic group is a numerical
invariant invented by
Buhler and Reichstein to measure the complexity of torsors for that group.
In this lecture, I will report on joint work with Reichstein and Vistoli
generalizing this notion to algebraic stacks.

Speaker:Tommaso de FernexTitle: Birational rigidity of hypersurfacesAbstract:
Pukhlikov conjectured that for N greater or equal to 4, all smooth
hypersurfaces of degree N in the N-dimensional projective space
nonrational in a very strong sense, namely, that they are birationally
superrigid, the case N=4 of this result being the celebrated theorem
of Iskovskikh and Manin. In this talk I will give a proof of
Pukhlikov's conjecture. After introducing the notion of birational
rigidity and motivating it from the minimal model program, I will
overview the methods (old and new) that are involved in the proof of
the conjecture. This will lead me to talk about multiplier ideals, and
to explan how arc spaces can be useful in order to study suitable
restriction properties of these ideals.

Speaker:Paul HackingTitle: Smoothable del Pezzo surfaces with quotient singularitiesAbstract:
We describe the classification of del Pezzo surfaces of Picard
rank one with smoothable quotient singularities. These include the well
known examples with A,D,E singularities, but there are many more where the
canonical divisor is not Cartier. There are several infinite series of
toric examples with a suprising combinatorial structure, and all but a
finite number of exceptional cases are obtained as deformations of the
toric examples if K^2 > 1. Joint work with Yuri Prokhorov.

Speaker:Max LieblichTitle: Boundedness of families of canonically polarized manifoldsAbstract:
In 1962, Shafarevich conjectured (among other things) that the
number of non-isotrivial families of proper smooth curves of genus g over
a fixed smooth base curve B is finite. This conjecture, which was proven
by Parshin (for proper B) and Arakelov (in general), has strong ties to
arithmetic (Mordell's conjecture) and to the geometry of the moduli space
of curves of genus g.
I will describe joint work with S\'andor Kov\'acs in which we prove a
generalization of this conjecture for families with higher-dimensional
fibers over bases of arbitrary dimension. The proof is independent of the
Minimal Model Program (although this is now perhaps irrelevant!).

Speaker:Sam PayneTitle: Toward foundations of tropical geometryAbstract:
Tropicalization, the degeneration of algebraic or analytic spaces to
piecewise linear objects, has recently emerged as a powerful technique
with applications to real and complex enumerative geometry, topology
of real algebraic varieties, and the arithmetic of heights. In this
talk, I will propose a definition of abstract tropical varieties as
thickenings of polyhedral complexes with integral structure, building
upon earlier ideas of Kempf, Knudsen, Mumford, and Saint-Donat, as
well as recent work of Kontsevich, Mikhalkin, Soibelman, and many
others.